1. Field of the Invention
A method for characterizing parameters describing microelectromechanical (MEMS) structures resulting from a device fabrication process or process variations is presented.
2. Description of Related Art
Monitoring the geometry of fabricated MEMS devices is as fundamentally important to predicting and evaluating device performance as tracking the material properties of the devices or variations in the device fabrication processes. Accurate knowledge and predictability regarding MEMS device geometries can improve device yields and reduce the need for post-fabrication processes such as trimming to achieve targeted device performance specifications. Although material properties generally affect critical performance parameters linearly, device geometry affects device performance characteristics to much higher powers, i.e., the critical performance characteristics of fabricated MEMS devices are generally much more sensitive to variations in device geometry than to variations in material properties. FIG. 6 summarizes mathematically the extent to which material properties and device geometry affect common critical performance parameters for several different MEMS devices. As can be seen, geometric parameters such as length (L) and width (w) affect critical performance parameters such as mechanical displacement or mechanical force by as much as the third or fourth power, whereas material properties such as density or modulus of elasticity generally only affect such parameters to the first power, i.e., linearly.
Unknown and undesired variations in device geometry can also adversely affect the methods used to extract material properties for MEMS devices from MEMS test structures. Typically, values of selected material properties are extracted from such test structures by physically measuring intermediate parameters, such as pull-in voltages, resonant frequencies, and maximum deflections. The measured values of the intermediate parameters can then be mathematically reduced to derive the desired material property values. However, the measured values of the intermediate parameters depend on device geometry and are sensitive to variations in fabricated geometric dimensions. It is desirable to eliminate such variations as much as possible so that data reduction from the intermediate parameters results in an accurate determination of the values of the selected material properties. The uncertainty introduced by variations in the fabricated geometric dimensions in such test structures is significant and introduces at least first-order errors in the inferred values of the material properties. Variations in geometric dimensions as large as 20% compared to nominal values can easily propagate to errors in extracted material property data as high as 100%.
Conventionally, measurements of MEMS geometries are accomplished ex-situ, and include the use of SEM""s, surface profilometers, ellipsometers, and interferometers. However, conventional ex-situ measurement techniques have various limitations and drawbacks. In some cases, measurements can be made to better than 5% accuracy, provided calibration standards for the measurement devices are carefully adhered to. This is particularly essential with respect to SEMs and surface profilometers. In addition, with a SEM, care is needed to align the viewing angle to be orthogonal to the surface of the device being measured, otherwise scaling offsets will be introduced in the dimensional measurements. Ellipsometers are specific to the substrates and layers used in device fabrication and are therefore of limited use. Further, ellipsometers require a laser spot size bigger than typical MEMS devices. Single-wavelength (xcex) interferometers may be useful for measuring certain geometric parameters. However, when depth measurements smaller than xcex/8 are required, errors can be introduced if the measurement techniques employed do not adjust for imperfect reflection off a device""s surface or for thin-film effects resulting from internal reflection within a region of varying thickness or a gap region underneath the device. This may happen with polysilicon and other materials commonly used to fabricate MEMS when they are fabricated less than 3 xcexcm thick and are partially transmissive at the optical wavelengths commonly used in interferometers.
In-situ electronic test structures for monitoring fabricated MEMS geometries and material properties are attractive because they offer ease of use, the repeatability and control of voltage application, compatibility with standard IC wafer-level probing techniques, limited device area requirements, and integration with real devices. In-situ measurement is especially useful for monitoring material properties which can be highly process-dependent. Mechanical property test structures using in-situ electrostatic actuation include beams and diaphragms actuated to pull-in, laterally resonant comb-drives, vertically resonant beams, and capacitance-voltage measurements of fixed-fixed beam bridges. Non-electrostatic methods for mechanical property measurement include load-deflection and bulge tests of membranes, measuring cantilever tip displacement with an externally applied force, and direct tensile measurement of strain. Non-electrostatic methods are typically carried out ex-situ, and require either specialized structures with modifications or additions to the fabrication process, or special apparatuses to make measurements and apply external forces.
Clearly, the ability to independently characterize the geometry of MEMS devices is essential to accurate, efficient, and successful device design, simulation, and material property extraction. Current methods of ex-situ and in-situ characterization have a variety of limitations and drawbacks. The present invention presents an approach for providing device geometry characterization which overcomes the limitations and drawbacks of the current methods and which advantageously relies only on an optical microscope and standard electronic test equipment used at the wafer-level for independent measurement of geometry.
Other work relevant to characterizing MEMS structures includes the following references, some of which are referred to by reference number in the following sections:
1. Raj K. Gupta, xe2x80x9cElectrostatic Pull-in Test Structure Design for Mechanical Property Characterization of Microelectromechanical Systems (MEMS)xe2x80x9d, Ph.D. Thesis, June 1997, M.I.T., Cambridge, Mass.
2. William C. Tang, xe2x80x9cElectrostatic Comb Drive for Resonant Sensor and Actuator Applicationsxe2x80x9d, Ph.D. Thesis, University of California at Berkeley, 1990.
3. R. I. Pratt, G. C. Johnson, R. T. Howe, and D. J. Nikkel, Jr., xe2x80x9cCharacterization of Thin Films Using Micromechanical Structuresxe2x80x9d, Materials Research Society Symposium Proceedings, 276 (1992) pp. 197-202.
4. H. Kahn, S. Stemmer, K. Nandakumar, A. H. Heuer, R. L. Mullen, R. Ballarini, and M. A. Huff, xe2x80x9cMechanical Properties of Thick, Surface Micromachined Polysilicon Filmsxe2x80x9d, Proceedings IEEE MEMS 1996, San Diego, Calif., Feb. 11-15, 1996, pp. 343-348.
5. M. Biebl, G. Brandl, and R. T. Howe, xe2x80x9cYoung""s Modulus of in-situ Phosphorus-doped Polysiliconxe2x80x9d, Proceedings of Transducers"" 1995, Volume II, Stockholm, SWEDEN, June 1995, pp. 80-83.
6. Jacob P. Den Hartog, xe2x80x9cMechanical Vibrationsxe2x80x9d, Fourth Edition reprint, Dover Publishing, Inc., Mineola, N.Y., USA, 1985, ISBN 0-486-64785-4.
7. M. A. Schmidt and R. T. Howe, xe2x80x9cSilicon Resonant Microsensorsxe2x80x9d, Ceramic Engineering and Science Proceedings, 8, No. 9-10, September-October 1987, pp. 1019-1034.
8. G. K. Fedder, S. Iyer, and T. Mukherjee, xe2x80x9cAutomated Optimal Synthesis of Microresonatorsxe2x80x9d, Proceedings of Transducers"" 97, Volume II, Chicago, Ill., USA, Jun. 16-19, 1997, pp. 1109-1112.
9. K. Wang and C. T.-C. Nguyen, xe2x80x9cHigh-Order Micromechanical Electronic Filtersxe2x80x9d, Tenth IEEE International Workshop on MEMS 1997, Nagoya, JAPAN, Jan. 26-30, 1997, pp. 25-30.
10. MEMCAD software is available from Microcosm Technologies, Inc., Cary, N.C.
11. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, Second Edition, Cambridge University Press, 1992, pp. 681-688.
12. S. R. Schmidt and R. G. Launsby, Understanding Industrial Designed Experiments, Air Academy Press, 1992.
The present invention advantageously addresses the need for independent, in-situ, wafer-level measurement of MEMS device geometry using electronic test structures. The approach of the invention is flexible and may be employed with a variety of known MEMS structures, including variations on the conventional comb-drive resonator structure, a solid model of which is shown in FIG. 9. The present invention avoids the drawbacks and limitations of known in-situ and ex-situ methods by exploiting the known sensitivity of readily measurable mechanical parameters of the chosen test structure to variations in the nominal values of selected geometric dimensions of the structure. This relationship can then be used to quantify unknown values of microscopic variations in the same geometric dimensions of similar fabricated devices. For example, the resonant frequency of the comb-drive resonator structure of FIG. 9 is known to be sensitive to certain geometric parameters such as sidewall angle xcex8 206 and edge bias woff 205, which are defined in FIG. 10. Similar physical parameters which are sensitive to other geometric parameters can be identified for other MEMS structures if desired. For the resonator structure shown in FIG. 9, the change in resonant frequencies due to typical manufacturing variations in sidewall angle xcex8 206 and edge bias woff 205 are shown in FIG. 12a and FIG. 12b. The relationship between geometric parameters and resonant frequencies are taught in Reference [6].
In the present invention, the unknown dimensional values of selected geometric parameters in fabricated MEMS structures can be extracted by first fabricating a number of similar test structures in which the dimensional values of the selected geometric parameters are varied by a known amount. For example, a number of comb-drive test structures can be fabricated with varying values of mask-drawn, i.e., nominal, geometric parameters such as width wo 309 and length L 308 (see FIGS. 7, 8, and 10). The resonant frequencies of the test structures can then be physically measured using known techniques. Known models are then applied to the observed resonant frequencies and are fit to the set of selected geometric parameters of unknown value in order to quantify them. The same approach may be used to quantify unknown values of selected material parameters or unknown values of process parameters. The fit assumes uniformity of processing of the test structures, thus eliminating as factors variations in a significant number of geometric parameters that could otherwise impact the outcome and avoiding the need for geometric calibration. For example, in the case of the comb-drive test structure, the fit assumes that the etched sidewall profiles, material and process are uniform across the devices measured. This is generally a suitable assumption in the case of similarly oriented devices fabricated on the same chip or wafer.
Resonant frequency models for the lateral natural frequency fH 505 of comb-drive MEMS structures have been investigated using analytical techniques and reported in References 2, 3, 4, 5, 6, 7, 8, and 9. The present invention extends these models by adding refinement with 3D analysis and a careful consideration of geometry. In particular, the extended models of the invention include the effects of (1) inertia and stiffness changes due to edge bias and sidewall angles, (2) compliant supports, (3) a distributed mass, (4) residual biaxial stress, and (5) the input of mask-drawn tether width variations on the preferred embodiment of a comb-finger MEMS structure.
As will be appreciated by those skilled in the art, the approach of the present invention is advantageous in that it only requires making prescribed changes in geometry at the photolithographic mask level rather than at the device level, and can be used with any MEMS structure which is mechanically sensitive to variations in geometric parameters and any suitable and accurate force-displacement measurement technique. The use of resonant frequency measurements is particularly advantageous and desirable since such measurements tend to limit nonlinear coupling between the drive force and mechanical response which can result from application of a strong drive. As is well known, for example, coupling of a strong electrostatic drive to the mechanics can introduce electromechanical xe2x80x9cspring softening,xe2x80x9d and in extreme cases causes pull-in, and in comb-drives can cause levitation for an in-plane drive.
The present invention provides a method and apparatus to characterize MEMS structures which can be used to determine the unknown values of device geometric parameters resulting from the device fabrication process or process variations. The methods and apparatus use controlled variations of selected geometric parameters of a single test structure design to obviate the need for independent geometric calibration, and the extraction approach employed allows measurements of geometry perturbations at a level of precision limited only by the minimum feature size of the fabrication process being used. The method and apparatus achieve a high degree of accuracy by using numerical analysis models which carefully consider geometry and residual stress.
In one preferred form of the invention, the nominal geometries of a MEMS structure, such as a resonant comb-drive, are selected such that a physically measurable performance parameter, such as the natural frequency fo of the device will fall within an experimentally measurable range.
A method of numerical analyses is used to accurately calculate the value of the selected performance parameter, e.g., natural frequency fo , of the structure from the nominal values of selected geometric parameters, such as thickness to, width wo, and gap go, and from selected independently measured mechanical properties, such as density xcfx81, modulus of elasticity E, residual stress "sgr"o, and poisson""s ratio xcexd. To calculate the natural frequency fo, for example, it is sufficient to know the ratio E/xcfx81. Also, knowing "sgr"o(1xe2x88x92xcexd) is sufficient to estimate changes in the natural frequency due to residual stress. Ideally the actual thickness t and gap g would be used to calculate natural frequency fo, but the error in not knowing the exact values is sufficiently minor that it may be ignored in most applications.
A set of photolithographic masks for a set of test structures are designed. The masks are designed to have varying trial values for the selected geometric parameters, the trial values being selected to encompass the entire range over which the unknown values of the selected geometric or material parameters as fabricated can be independently monitored. This range typically should include the smallest feature width w that can be fabricated using the lithography process.
The test structures are fabricated and a set of experimentally observed values for the selected performance parameter, e.g. a set of natural frequencies, is obtained. For example, observed frequencies may include lateral natural frequency fL, lateral natural frequency fH, and torsional natural frequency fT, which are all dependent on the selected trial values. This set of experimentally observed performance parameter values, e.g., natural frequencies, is compared with the calculated values of models described by the trial values. Minimizing the error measure between the experimentally observed values and calculated values results in high-accuracy numerical approximations of the unknown parameters.
A preferred form of the present invention utilizes the lateral resonant mode in the MEMS comb-drive resonator 102 based on the mask layout in FIG. 7. Not only does the lateral resonant mode have a large sensitivity to process description parameters edge bias woff 205 and sidewall angle xcex8 206 (see FIG. 12a and FIG. 12b), but this mode is easy to experimentally excite and to detect. In addition the lateral resonant mode is first-order insensitive to fabricated thickness variations, especially for small sidewall angles.
The comb-drive resonator 102 is suitable to illustrate a preferred form of the invention because of its wide-spread use in the MEMS community and because it demonstrates how the geometric extraction technique of the invention can be applied to fabricated parts. However, the present invention is not limited to use with the particular comb-drive resonator structure illustrated herein and is sufficiently general to be applied to most geometric design variations of the comb-drive resonators presented in literature, as well as other MEMS structures, including designs for surface micromachining, LIGA, crystal-silicon wafer bonding, and bulk micromachining.